The covariant Gaussian approximation is applied to a constrained theory-the 2-dimensional O(N) nonlinear σ-model. The application of Gaussian approximation to constrained systems is discussed. The U(1) gauge symmetry discussed earlier in the framework of this model exhibits itself on the quantum level. Several features common with 4-dimensional gauge theories are clearly seen on this simple example. It is found that all the known qualitative features of the model are correctly reproduced in this approximation. These include dynamical mass generation, asymptotic freedom, dimensional transmutation, factorization of the S-matrix.